Probabilistic trace and Poisson summation formulae on locally compact abelian groups

David Applebaum
2017 Forum mathematicum  
We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the
more » ... ociated semigroup on L 2 -space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula. MSC 2010: Primary 60B15, Secondary 60E07, 11F85, 43A25, 11R56.
doi:10.1515/forum-2017-0049 fatcat:wce22xhxpfeojlaj2wpxaijoly